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Keywords
(10)
bayesian estimator
Error Bound
Estimation Error
Indexing Terms
Mathematical Model
Minimum Mean Square Error
Noise Reduction
Signal Processing
Sparse Representation
Upper Bound
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On MMSE and MAP Denoising Under Sparse Representation Modeling Over a Unitary Dictionary
On MMSE and MAP Denoising Under Sparse Representation Modeling Over a Unitary Dictionary,10.1109/TSP.2011.2151190,IEEE Transactions on Signal Processi
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On MMSE and MAP Denoising Under Sparse Representation Modeling Over a Unitary Dictionary
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Javier S. Turek
,
Irad Yavneh
,
Michael Elad
Among the many ways to model signals, a recent ap proach that draws considerable attention is
sparse representation
modeling. In this model, the signal is assumed to be generated as a random linear combination of a few atoms from a prespecified dictionary. In this work, two Bayesian denoising algorithms are analyzed for this model—the maximum a posteriori probability (MAP) and the minimummeansquarederror (MMSE) estima tors, both under the assumption that the dictionary is unitary. It is well known that both these estimators lead to a scalar shrinkage on the transformed coefficients, albeit with a different response curve. We derive explicit expressions for the estimationerror for these two estimators. Upper bounds on these errors are developed and tied to the expected error of the socalled oracle estimator, for which the support is assumed to be known. This analysis establishes a worstcase gainfactor between the MAP/MMSE estimation errors and that of the oracle. Index Terms—Bayesian estimation, error bound, maximum a posteriori probability (MAP), minimummeansquarederror (MMSE), oracle, shrinkage, sparse representations, unitary dictionary.
Journal:
IEEE Transactions on Signal Processing  TSP
, vol. 59, no. 8, pp. 35263535, 2011
DOI:
10.1109/TSP.2011.2151190
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